Solutions to Assignment 3
SYSC 3501: Communication Theory, Summer 2016
Department of Systems and Computer Engineering
Carleton University
1. Problem 5.2
1
2. Problem 5.3
3. Problem 5.12
2
4. Problem 5.13
5. Problem 5.16
3
6. Problem 5.17
4
7. Problem 5.20

SYSC3501 Communication Theory
Assignment #1
(Summer 2011)
Problem 1: Determine the Fourier transform of the signal:
s (t ) 3 cos( 2000 t ) 5e |t |
Problem 2: Express the following periodic waveform using the complex Fourier series.
s(t)
2
1
-6
-5
-3
-2
1

SYSC3501 Communication Theory
Assignment #2
Due: 4:00PM, June 6, 2011
Question 1
An AM-USSB transmitter modulates a rectangular pulse, m(t ) = (t / 3) at a frequency
1 2t + 3
of 1000 Hz. A table of Hilbert transforms tells us that m(t ) = ln
.
2t 3
(a) F

SYSC3501 Communication Theory
Assignment #3
Due: 4:00PM, June 14, 2011
Question 1
The following digital baseband message is transmitted:
m(t )
1
1
1
0
1
0
t
2s
-1
(a) What is the pulse function p(t) ? What is its spectrum ?
(b) Assume BPSK modulation with

SYSC3501 Communication Theory
Assignment #4
Due: 4:00PM, June 20, 2011
Question 1
(a) Given the function 3 2, assume p(t) is ideally sampled at a rate of 4
Hz to produce the signal p(t). Sketch p(t) for -3 t 3. Will aliasing occur? Explain
why or why not.

SYSC 3501 Winter 2012
Solutions to Assignment 2
Problem 3 [10 Marks]
Consider the system shown below, where the input signals, x1(t) and x2(t), both have a bandwidth
of W Hz. If the lowpass filter also has a bandwidth of W Hz, and fc >W, express z(t) in t

SYSC3501 Communication Theory
Assignment #2
Due: 12:00pm Friday February 12
Problem 1 [10 Marks]
An AM-DSB-C transmitter is modulated with an audio testing signal given by
m(t ) = 0.2 sin 1t + 0.5 cos 2 t , where f1 =500Hz and f2= 500 2 Hz.
(a) Sketch the

SYSC3501 Communication Theory
Assignment #1
Due Jan 29, 2010
Assignment Bin is by the mens washroom near the SYSC office
(Winter 2010)
Problem 1[text 2.30]: Determine the Fourier transform of the signal:
s (t ) 3 cos( 2000 t ) 5e |t |
Problem 2 [text 2.13

SYSC3501 Communication Theory
Assignment #1
Solution Set
(Winter 2012)
Problem 1 : Determine the Fourier transform of the signal:
s(t ) 3 cos(2000 t ) 5e |t|
There are 2 approaches to answer this question. Both yield equivalent answers.
1st approach:
2nd

SYSC3501 Communication Theory
Assignment #3
Due: 4:00pm March 18th 2010
Question 1
Suppose a signal m(t) (shown in the figure below) is transmitted using frequency
modulation.
m(t)
2
0
1
2
3
4
t (sec)
-2
(a) Write the expression of fi(t), the instantaneou

SYSC3501 Communication Theory
Assignment #4
Due 4:00pm Friday April 9th 2010
Question 1
A binary waveform of 9,600 bits/s is converted into an 8-level waveform that is passed
directly through a channel using a raised cosine pulse. The channel has a freque

SYSC3501 Communication Theory
Assignment #3
Due: 4:00pm Friday March 16th 2012
Question 1
Suppose a signal m(t) (shown in the figure below) is transmitted using frequency
modulation.
m(t)
2
0
1
2
3
4
t (sec)
-2
(a) Write the expression of fi(t), the insta

SYSC3501 Communication Theory
Assignment #4
Due 4:00pm Wednesday April 4th 2012
Question 1
A binary waveform of 9,600 bits/s is converted into an 8-level waveform that is passed
directly through a channel using a raised cosine pulse. The channel has a fre

SYSC3501 Communication Theory
Assignment #2
[Due: 4:00pm Monday February 13th]
Problem 1 [10 Marks]
An AM-DSB-C transmitter is modulated with an audio testing signal given by
m(t ) = 0.2 sin 1t + 0.5 cos 2 t , where f1 =500Hz and f2= 500 2 Hz.
(a) Sketch

Comparison of Line Codes
Polar NRZ Signal
Advantages:
- Relatively easy to generate the signal but requires dual
supply voltages.
- Bit error probability performance is superior to other line
encoding schemes.
Disadvantages:
It has a large power spectral

Delta Modulation
Extreme version of DPCM is called delta modulation.
Difference between current sample and previous quantized
sample is encoded by one bit.
If m(nTs) m (n1)Ts) > O, transmitted bit = 1
If m(nT5) m (n1)Ts) < O, transmitted bit = 0
At the re

Intersymbol Interference
- If the rectangular multilevel pulses are filtered improperly as they pass
through a communications system, they will spread in time, and the pulse
for each symbol may be smeared into adjacent time slots and cause
Intersymbol Int

Pulse Code Modulation (PCM)
- The Analog-to-digital Converter(ADC)
performs three functions: Analog / \ A
Sampling: Makes the signal discrete Input 1 l l
in time. If the analog input has a Signal -
bandwidth of B Hz, then the
minimum sample frequency suc

Comparlson 01' Line C0065
Unipolar NRZ Signal
Advantage:
Relatively easy to generate the signal from a single
power supply.
Disadvantages:
A dc component is always present corresponding to a
waste of transmission power.
It has a large power spectral de

Quantization
The output of a sampler is still continuous in amplitude.
The number of possible values is infinite.
To transmit as a digital signal, we must restrict the number of
possible values.
Quantization is the process of "rounding off a sample accord

Differential PCM
In PCM, the consecutive samples are encoded independently.
In differential pulse code modulation (DPCM), it is the difference
between the current sample and the previous quantized sample
that is encoded.
When we sample at the Nyquist rate

Comparison OT Line COdES
Bipolar signaling has several advantages:
its spectrum has a dc null.
its bandwidth is not excessive.
it has singleerrordetection capability. This is a due to the fact
that if a single detection error is made, it will violate t

Intersymbol Interference (ISI)
Intersymbol interference (ISI) occurs when a pulse spreads
out in such a way that it interferes with adjacent pulses at
the sample instant.
Example: assume polar NRZ line code. The channel outputs
are shown as it spreads due

Combating ISI
Three strategies for eliminating ISI:
Use a line code that is absolutely bandlimited.
Would require Sinc pulse shape.
- Cant actually do this (but can approximate).
Use a line code that is zero during adjacent sample instants.
- Its okay

Comparison of Line Codes
Bipolar RZ Signal
Advantages:
- There is a null at dc so that an accoupled circuit may be used in
the transmission path.
It has singleerrordetection capability since a single error will
cause a violation (the reception of 2 or mo

Pulse-Code Modulation
Benefits of PCM
PCM is reliable, inexpensive, and highly resistant to noise. The noise
performance of a digital system can be superior to that of an analog
system.
The transmitted binary pulses all have the same amplitude and can be

Comparison 01' LIne COdes
Unipolar RZ Signal
Advantage:
Relatively good clock recovery strings of 1s no longer cause
problem.
Disadvantages:
The first null bandwidth is twice that for the polar NRZ signal
or the unipolar NRZ signal.
A discrete impulse

Question 1 (30%)
Part Ib) (5%) Assume a signal
4
met) = 1O[Sa(1rI0 t)]2 that is fed into an FM modulator. The
J
resulting modulated signal is sCt) =cos(21r108 t+80001r m(t)dt). Write an expression for the
instantaneous frequency of the modulated wave. W